Axisymmetric waves in electrohydrodynamic flows

The formation of nonlinear axisymmetric waves on inviscid irrotational liquid jets in the presence of radial electric fields is considered. Gravity is neglected but surface tension is considered. Electrohydrodynamic waves of arbitrary amplitude and wavelength are computed using finite-difference methods. Particular attention is paid to nonlinear traveling waves. In the first class of problems, an electric field generated by placing the liquid jet inside a hollow cylindrical electrode held at constant voltage, its axis coinciding with that of the jet, is studied. The jet is assumed to be a perfect conductor whose free surface is stressed by the electric field acting in the hydrodynamically passive annulus. In the second class of problems, the annular gas is a perfect conductor that transmits a constant voltage onto the liquid/gas surface. The liquid axisymmetrically wets a constant-radius cylindrical rod electrode placed coaxially with respect to the hollow outer electrode, and held at a different constant voltage. The fluid dynamics and electrostatics need to be addressed simultaneously in the inner region. Axisymmetric interfacial waves influenced by surface tension and electrical stresses are computed in both cases. The computations are capable of following highly nonlinear solutions and predict, for certain parameter values, the onset of interface pinching accompanied with the formation of toroidal bubbles. For given wave amplitudes, the results suggest that, for the former case, the electric field delays bubble formation and reduces wave steepness, while for the latter case the electric field promotes bubble formation, all other parameters being equal.     

Axisymmetric delaminations in layered cylinders

An analytical model is developed to predict the effect of an axisymmetric delamination in a layered anisotropic cylinder under different loading conditions. Thin shell assumptions are incorporated into classical laminated shell theory to facilitate the use of the simplified Donnell equations for shell displacements. The critical loading condition under which the debond tends to propagate is predicted based on Griffith's energy balance criterion. A closed-form solution for a simplified model is shown to form the basis for the realistic problem. The effects of the individual ply thicknesses, the crack front curvature, and the axial and shear strains of the laminate reference plane are studied.

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Résumé Un modèle analytique est développé pour prédire l'effet d'une délamination axisymétrique dans le cas d'un cylindre anisotrope à plusieurs couches sous des conditions de charge différentes. On incorpore des hypothèses sur les coques minces dans la théorie des coques colaminées classiques afin de faciliter l'utilisation des équations simplifiées de Donnell pour les déplacements de la coque. La condition critique de sollicitation sous laquelle le décollement tend à se propager est prédite sur la base du critère d'équilibre d'énergie de Griffith. Une solution fermée pour un modèle simplifié apparait constituer la base d'un problème réaliste. Les effets d'épaisseur des différentes couches, la courbure du front de la fissure et les déformations axiales et de cisaillement du plan de référence de la liaison sont étudiés.

     

Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and multiquadrics discretizations

In this paper, the static and free vibration analysis of composite plates are performed, using a layerwise deformation theory and multiquadrics discretization. This meshless discretization method considers radial basis functions as the approximation method for both the equations of motion and the boundary conditions. The combination of this layerwise theory and the multiquadrics discretization method allows a very accurate prediction of the natural frequencies.     

Axisymmetric oscillations of a cupola-shaped shell

A spectral boundary problem on axisymmetric eigenoscillations of a cupola-shaped shell is considered with emphasis on small shell thickness. The problem deals with a singularly perturbed system of ordinary differential equations. The paper examines analytical properties of the solution and, based on that, constructs an appropriate functional basis for Ritz’ method. Employing this basis provides fast convergence in the C ³-metrics.     

Axisymmetric fretting analysis in coated cylinder

Fretting is essentially a contact fatigue phenomenon, although bulk stresses and material properties contribute to final failure. The near surface state of stress developed under oscillatory contact between machine elements plays a major role in deciding the severity of fretting. It is possible to enhance tribological properties by coating the surface. There is rather scanty literature available on fretting analysis of coated components. Presence of such coatings has a large influence on the near surface state of stress. The effect of coatings on the severity of fretting is the focus of this paper. Results obtained for both hard and soft coatings are compared with the results obtained for the homogeneous case. The component geometry and loading are chosen to be cylindrical to enable 3D elastic axisymmetric fretting analysis. The results are compared with 2D models (strip and half-plane) to examine their utility and validity for understanding axisymmetric fretting. Contact pressure and frictional shear loading cases are solved separately and superposed appropriately depending on the coefficient of friction considered. Results for different values of coefficient of friction and elastic mismatch are illustrated through contour plots of stresses and strains. These results are expected to be helpful for identifying fretting failure zones and fracture mechanisms in coated components. Analytical results presented here could serve as useful benchmarks for calibrating numerical codes and experimental techniques.     

Axisymmetric Contact Problem for an Interface Crack

We study stresses formed in two rigidly connected elastic half spaces made of different materials in the case where their interface contains a circular crack. The contact of the crack surfaces along the crack contour is taken into account and the solution of the integral equation of the problem is obtained by the Wiener–Hopf method in the closed form. The size of the zone of contact of the crack surfaces and the distributions of stresses in the contact region and on the interface of the half spaces outside the crack are found explicitly.     

Axisymmetric crack with a slightly non-flat surface

A three-dimensional axisymmetric crack with a slightly non-flat surface in an isotropic linear solid under axisymmetric loading is analyzed. The problem is formulated by using the Hankel transform and a perturbation solution is obtained, which is accurate to the first order of the parameter representing the non-flatness. The stress intensity factor for the problem is evaluated. In particular, the stress intensity factor at the onset of axisymmetric kinking from a penny-shaped crack is obtained. It is also shown that the two-dimensional Cotterell-Rice theory for the effect of tensile stress acting parallel to the crack surface on the stability of crack path is valid for the axisymmetric crack.     

Axisymmetric postbuckling of orthotropic thick annular plates

Summary This study deals with the geometrically nonlinear postbuckling response of cylindrically orthotropic thick annular plates subjected to inplane radial compressive load at the outer edge. Clamped and simply supported plates with a free hole and a plugged hole have been considered. Governing differential equations are expressed in terms of transverse displacementw, shear rotation and stress function . Polynomial expansions are employed for these three field variables and the orthogonal point collocation method has been used to obtain the discretised equations. Buckling loads have been determined by solving a linear eigen-value problem using the method of inverse iterations. Postbuckling loads for different central deflections have been obtained by solving the nonlinear differential equations. The effect of thickness ratio, orthotropic parameter, annular ratio and the edge conditions has been investigated. The effect of shear deformation on the buckling loads increases with the orthotropic parameter and the thickness-to-radius ratio.Notations a, b, h Outer and inner radii and thickness of the plate - E ,E ,v ,v Young's moduli, Poisson's ratios - Orthotropic parameter:E /E _r - G _rz /E _r - u ^*,u ₀ ^* Radial displacement, radial displacement at mid plane - r, , z Cylindrical polar coordinates - , ^* Strains and stresses - w ^*,w ,w Transverse displacement, shear rotation and stress function - w, , Dimensionless deflection, shear rotation and stress function - N _r ,N ,Q _r Inplane stress resultants and transverse shear - M _r ,M Moment resultants - k ² Shear correction factor - , (r–b)/(a–b), b/(a–b) - a/h - _r ^b , ^m Radial bending stress, circumferential membrane stress - N, _i Number and radii of collocation points - Eigenvalue for buckling With 6 Figures     

Axisymmetric analytical stiffness matrices with Green-Lagrange strains

Stiffness matrices based on the non-linear Green-Lagrange definition seem complicated, but for the case of a linear displacement ring-element with triangular cross-section, closed form final results are listed, directly suited for coding in a finite element program. These analytical secant and tangent element stiffness matrices are obtained by separating the dependence on the material constitutive parameters and on the stress/strain state from the dependence on the initial geometry and the displacement assumption. As an example of application, numerical results for a circular plate problem show the indirect severe errors that may result from a linear strain model. It is difficult to predict the indirect errors that follow from the erroneous displacement field, and the explanations behind such predictions are attempted. The nodal positions of an element and the displacement assumption give six basic matrices that do not depend on material and stress strain state, and thus are unchanged during the necessary iterations for obtaining a solution based on Green-Lagrange strain measure. The presented resulting stiffness matrices are especially useful in design optimization, because analytical sensitivity analysis can then be performed.     

Axisymmetric elastodynamic infinite elements for multi-layered half-space

This paper presents three axisymmetric infinite elements for the elastodynamic problems in a multi-layered half-space. They are the horizontal, the vertical and the corner infinite elements that are developed by using the wave functions in the function spaces derived from approximate expressions of the analytical solutions. An efficient integration procedure is proposed for calculating the element matrices involving multiple wave components. Numerical example analyses are presented for rigid disks on homogeneous and layered half-spaces, and for an embedded caisson. The numerical results obtained show the effectiveness of the proposed infinite elements.     

Axisymmetric vibration of piezocomposite hollow circular cylinder

Summary The axisymmetric vibration of a piezoelectric laminated hollow circular cylinder has been studied for an imperfect interface model. The frequency equation is derived for traction free inner and outer surfaces of the hollow cylinder with continuity conditions at the bonding interfaces. The composite cylinder is composed of two different piezoelectric materials belonging to 6 mm class and a hypothetical Linear Elastic Material with Voids (LEMV) as bonding layer. Numerical solutions of the frequency equation are obtained for the composite cylinder ceramic(1)/LEMV/ceramic(2). Computational results are presented as dispersion curves as well as in tables to characterize the attenuation of axial waves for three layered cylinders with and without voids in the thin LEMV layer.     

Two-Level Finite Element Study of Axisymmetric Cracks

We extend the two-level finite element method (2LFEM) to the accurate analysis of axisymmetric cracks, where both the crack geometry and applied loads are symmetrical about the axis of rotation. The complete eigenfunction expansion series for axisymmetric cracks developed by us are employed as the global interpolation function such that the stress intensity factors are primary unknowns. The coupled coefficients in the series are solved iteratively. The stress intensity factors are computed directly from the coefficients for any arbitrary axisymmetric loading on the boundary. Engineering applications of 2LFEM to numerical fracture mechanics analysis for stress intensity factors include several examples: Penny-shaped and circumferential cracks in round bars; and internal and edge circumferential cracks in thick wall pipes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Gravity Effect on the Axisymmetric Drop Spreading

The flattening (spreading) of the axisymmetrical drop on a plane horizontal surface under action of gravity force at zero tangential force (no shear at the gas–liquid interface) is investigated analytically and numerically. We determine the exact profile of compressed drop assuming the condition of drop volume conservation. 2D time dependant numerical model, based on a finite difference method, has been developed to describe the hydrodynamics inside the drop. The energy and Navier–Stokes equations are solved within the drop’s analytical profile. Effects of surface tension and thermocapillarity are taken into account. The effect of gravity has been studied to define main features of the drop dynamics. In calculations vector of gravitational acceleration is oriented perpendicularly to the surface, the Bond number is changed in the range from Bo = 0 to Bo = 151.6. Our results show that the gravity has a significant effect on the drop spreading.     

Axisymmetric graph model of an elastic solid

The method of a deformable solid analysis based on graph theory principles is used in mechanics to construct discrete models that allow a numerical simulation of the solid displacement, strain, and stress fields. It provides the alternative opportunity to derive solid equations by transformation of the generalized coordinate systems introduced by the graph structural objects (paths, cut-sets, cycles, chords).Published in Problemy Prochnosti, No. 6, pp. 83–103, November–December, 1996.     

Propagation of Substance in an Axisymmetric Filtration Flow

Journal of Engineering Physics and Thermophysics - The article considers the propagation of a substance in a cylindrical coordinate system in filtration flow. A common assumption is made that the...     

Axisymmetric viscous liquid oscillations in a cylindrical container

The axisymmetric natural damped frequencies (m=0) of a viscous liquid in a cylindrical container are obtained for slipping and anchored contact line at the container wall. The results may also be applied to viscous liquid in a micro gravity environment, if the contact angle of the contact line to the container wall is in the vicinity of π/2, indicating that the free liquid surface remains a plane surface in equilibrium. It was found that there exists, in contrast to frictionless liquid, a range where only aperiodic motion of the liquid is possible. This appears for small liquid height ratios h/a.     

Axisymmetric deformation of a short magneto-strictive current-carrying cylinder

The title-problem has been reduced to that of solving a Fredholm integral equation of the second kind. One end of the cylinder is assumed to be fixed, while the cylinder is deformed by an axial current. The vertical displacement on the upper flat end of the cylinder has been determined from an iterative solution of the Fredholm equation valid for large values of the length. The radial displacement of the curved boundary has also been determined at the middle of the cylinder, by using the iterative solution.